On a combinatorial problem of Asmus Schmidt (Q1883632)

Summary: For any integer \(r\geq 2\), define a sequence of numbers \(\{c^{(r)}_k\}_{k=0,1,\dots}\), independent of the parameter \(n\), by \[ \sum^n_{k=0}\binom nk^r\binom{n+k}{k}^r=\sum^n_{k=0}\binom nk\binom{n+k}{k}c^{(r)}_k,\quad n=0,1,2,\dots. \] We prove that all the numbers \(c^{(r)}_k\) are integers.